KRM - Benchmark, fluid sphere and sardine

Testing the KRMr implementation of the KRM - Kirchhoff Ray Mode - model, based on equations described in detail in ‘Acoustic models of fish: The Atlantic cod (Gadus morhua)’ by Clay and Horne, 1994.

The KRM and the underlying scattering theory holds validity for high frequencies (>>> resonance freqeuncy) and for incident angles between 65 and 115 degrees.

Shapes are approximated shapes through a series of cylinders. x coordinates are the along body coordinates (major axis of an ellipsoid) z coordinates are the upper and lower coordinates at position x (the upper and lower values of the minor axis). w defines the width frpm a top view along the body axis.

The model is split in two parts:
1) The Swimbladder (or gas bubble) part
2) The Fish body (or fluid like) part

Weakly scattering fluid Sphere

To approximate scattering by a fluid sphere through the KRM mo9del, we set the swimbladder part to 0 and only consider the Fish body part.
We first define the sphere parameters:

#library(KRMr)
#Define the the sphere parameters
freqs = 10:400 * 1000 #Hz frequencies to be considered in Hz
a = 0.01 #m radius of the sphere
cw = 1477.4 #m/s sound speed surrounding water
cfb =  1480.3 #m/s soundspeed inside the sphere
rhow = 1026.8 #kg/m3 density surrounding water
rhofb = 1028.9 #kg/m3 density fluid inside sphere

Now we can construct shape information that is understood by the KRM model:

#generate sphere coordinates
ang = seq(-90,90,length.out=30) #angles to consider
shape=data.frame(
  xfb = sin(ang * pi / 180), #x coordinates of a sphere
  wfb = (2*cos(ang * pi / 180)), #diameter of the sphere
  zfbU = cos(ang * pi / 180), #upper z coordinates
  zfbL = -cos(ang * pi / 180)) #lower z coordiates

shplot(x_fb = shape$xfb,w_fb = shape$wfb,z_fbU = shape$zfbU, z_fbL=shape$zfbL)

Plot in 3D:

get_shp3d(shape)
#> 2023-03-06 18:50:21:1 Body parts detected

Now we can run the KRM simulation:

kfs = krm(frequency =freqs,
              c.w = cw,
              rho.w = rhow,
              theta=90,
              cs = cfb,
              rhos = rhofb,
              modes="fluid",
              L=0.02,
              shape=shape)
#> 2023-03-06 18:50:21:1 Body parts detected with 1 densities and 1 internal soundspeeds

We will use the analytical solution of the scattering by a fluid sphere as descirbed in Jech et al. 2015, modified from Anderson (1950).

Jech, J. M., Horne, J. K., Chu, D., Demer, D. A., Francis, D. T., Gorska, N., … & Reeder, D. B. (2015).”Comparisons among ten models of acoustic backscattering used in aquatic ecosystem research.” The Journal of the Acoustical Society of America, 138(6), 3742-3764.
Anderson, V. C. (1950). “Sound scattering from a fluid sphere.” The Journal of the Acoustical Society of America, 22(4), 426-431.

kfs$bench=sapply(freqs,
                 function(f){
                   fsphere(f,a,cw,cfb/cw,
                             rhofb/rhow)})

Plotting the results:

colors <- c("Analytical" = "red", "KRM" = "blue")

ggplot2::ggplot()+
  ggplot2::geom_line(data=kfs, ggplot2::aes(x=frequency/1000,y=TS, group=L,col='KRM'),lwd=1)+
  ggplot2::geom_line(data=kfs, ggplot2::aes(x=frequency/1000, group=L, y=bench,col='Analytical'), lty=2,lwd=1)+
  ggplot2::scale_color_manual(values = colors, name='')+
  ggplot2::xlab('Frequency (kHz)')+
  ggplot2::ylab('Target Strength (dB re m-2)')+
  ggplot2::theme_classic()+
  ggplot2::theme(text=ggplot2::element_text(size=16),
        legend.position='top')

Gas filled fluid sphere


#Define the the sphere parameters
freqs = seq(12,400,1) * 1000 #Hz frequencies to be considered in Hz
a = 0.01 #m radius of the sphere
cw = 1477.4 #m/s sound speed surrounding water
cfb =  345.0 #m/s soundspeed inside the sphere
rhow = 1026.8 #kg/m3 density surrounding water
rhofb = 1.24 #kg/m3 density fluid inside sphere

Now we can run the KRM simulation:

kfs = krm(frequency =freqs,
              c.w = cw,
              rho.w = rhow,
              theta=90,
              cs = cfb,
              rhos = rhofb,
              L=0.02,
              modes="soft",
              shape=shape)
#> 2023-03-06 18:50:24:1 Body parts detected with 1 densities and 1 internal soundspeeds

We will use the analytical solution of the scattering by a fluid sphere as descirbed in Jech et al. 2015, modified from Anderson (1950).

kfs$bench=sapply(freqs,
                 function(f){
                   fsphere(f,a,cw,cfb/cw,
                             rhofb/rhow)})

Plotting the results:

colors <- c("Analytical" = "red", "KRM" = "blue")

ggplot2::ggplot()+
  ggplot2::geom_line(data=kfs, ggplot2::aes(x=frequency/1000,y=TS, group=L,col='KRM'),lwd=1)+
  ggplot2::geom_line(data=kfs, ggplot2::aes(x=frequency/1000, group=L, y=bench,col='Analytical'), lty=2,lwd=1)+
  ggplot2::scale_color_manual(values = colors, name='')+
  scale_y_continuous(limits=c(-50,-40))+
  ggplot2::xlab('Frequency (kHz)')+
  ggplot2::ylab('Target Strength (dB re m-2)')+
  ggplot2::theme_classic()+
  ggplot2::theme(text=ggplot2::element_text(size=16),
        legend.position='top')

Sardine (like)

First we define the shape of a sardine like fish. Here we take the sardine shape profile from the NOAA SWFSC website.

#Sardine
data(pil_fb) #fish body sardine
fb=pil_fb
data(pil_sb) #swimbladder sardine
sb=pil_sb

Now we can have a look at the shape:

shplot(x_fb = fb$x_fb, w_fb = fb$w_fb, x_sb = sb$x_sb, w_sb = sb$w_sb,z_fbU = fb$z_fbU,z_fbL = fb$z_fbL,z_sbU = sb$z_sbU,z_sbL = sb$z_sbL)

get_shp3d(list(pil_fb, pil_sb))
#> 2023-03-06 18:50:26:2 Body parts detected

Because we have defined the fishbody coordinates and the swimbladder coordinates in a dataframe, the most simple way to run one KRM simulation in KRMr is:

krm(shape = list(fb, sb))
#> 2023-03-06 18:50:29:2 Body parts detected with 2 densities and 2 internal soundspeeds
#>   frequency  c.w rho.w theta    L scale                        f1
#> 1    120000 1490  1030    90 0.21     1 -0.001188948-0.000358712i
#>                         f2        sigma       TS
#> 1 0.001346062+0.000360853i 0.0001571285 -76.0749

Now we are ready to run a KRM simulation. Let’s have a look at the TS for a 21 cm sardine from 120 to 200 kHz. Here we will define the different coordinates separately and add the parameters we want to simulate:

TS = krm(frequency =seq(20,200.1) * 1000,
         c.w = 1490,
         rho.w = 1030,
         theta=90,
         cs = c(1570, 345),
         rhos = c(1070,1.24),
         L=0.21,
         shape=list(fb,sb))
#> 2023-03-06 18:50:29:2 Body parts detected with 2 densities and 2 internal soundspeeds

Plot the results:

plot(TS$frequency/1000, 
     TS$TS, 
     type='l',
     xlab='Frequency (kHz)', 
     ylab=expression(TS~'('~dB~re~ m^-2~')'))

We can also run a simulation over multiple parameters, for example frequencies from 120-200 kHz and incident angles from 65-115 degrees:

TS = krm(frequency =seq(120,200.1) * 1000,
         c.w = 1490,
         rho.w = 1030,
         theta=65:115,
         cs = c(1570, 345),
         rhos = c(1070,1.24),
         L=0.21,
         shape=list(fb,sb))
#> 2023-03-06 18:50:31:2 Body parts detected with 2 densities and 2 internal soundspeeds

We can for example use ggplot2 to plot a 2D map of the results:

ggplot2::ggplot(data=TS, ggplot2::aes(x= frequency/1000, y= theta, fill=TS))+
  ggplot2::geom_tile()+
  ggplot2::scale_fill_viridis_c(expression(TS~'('~dB~re~ m^-1~')'))+
  ggplot2::xlab('Frequency (kHz)')+
  ggplot2::ylab(expression(theta~'(°)'))+
  ggplot2::scale_x_continuous(expand=c(0,0))+
  ggplot2::scale_y_continuous(expand=c(0,0))+
  ggplot2::theme_classic()+
  ggplot2::theme(legend.position='top',
        legend.text=element_text(angle=45),
        text=element_text(size=16))

KRM - Benchmark, fluid sphere and sardine

Sven Gastauer

2023-03-06

Weakly scattering fluid Sphere

Gas filled fluid sphere

Sardine (like)